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A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow

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A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow AUGUST 2008 HARNIK ET AL. 2615 A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow N. HARNIK AND E. HEIFETZ Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, Israel O. M. UMURHAN Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, and Department of Physics, The Technion, Haifa, Israel, and City College of San Francisco, San Francisco, California F. LOTT Laboratoire de Meteorologie Dynamique, Ecole Normale Superieure, Paris, France (Manuscript received 4 September 2007, in final form 19 November 2007) ABSTRACT Motivated by the success of potential vorticity (PV) thinking for Rossby waves and related shear flow phenomena, this work develops a buoyancy–vorticity formulation of gravity waves in stratified shear flow, for which the nonlocality enters in the same way as it does for barotropic/baroclinic shear flows. This formulation provides a time integration scheme that is analogous to the time integration of the quasigeo- strophic equations with two, rather than one, prognostic equations, and a diagnostic equation for stream- function through a vorticity inversion. The invertibility of vorticity allows the development of a gravity wave kernel view, which provides a mechanistic rationalization of many aspects of the linear dynamics of stratified shear flow. The resulting kernel formulation is similar to the Rossby-based one obtained for barotropic and baroclinic instability; however, since there are two independent variables—vorticity and buoyancy—there are also two indepen- dent kernels at each level. Though having two kernels complicates the picture, the kernels are constructed so that they do not interact with each other at a given level. 1. Introduction ter (Drazin and Reid 1981). These conditions are quite easily obtained from the equations governing each type Stably stratified shear flows support two types of of instability, but their physical basis is much less clear. waves and associated instabilities—Rossby waves that We do not have an intuitive understanding as we do, for are related to horizontal potential vorticity (PV) gradi- example, for convective instability, which arises when ents, and gravity waves that are related to vertical den- the stratification itself is unstable (e.g., Rayleigh– sity gradients. Each of these wave types is associated Bernard and Rayleigh Taylor instabilities). with its own form of shear instabilities. Rossby wave There are two main attempts to physically under- instabilities (e.g., baroclinic instability) arise when the stand shear instabilities, which have gone a long way PV gradients change sign (Charney and Stern 1962), toward building a mechanistic picture. Overreflection while gravity wave–related instabilities, of the type de- theory (e.g., Lindzen 1988) explains perturbation scribed for example by the Taylor–Goldstein equation, growth in terms of an overreflection of waves in the arise in the presence of vertical shear, when the Rich- cross-shear direction, off of a critical level region. Un- ardson number at some place becomes less than a quar- der the right flow geometry, overreflected waves can be reflected back constructively to yield normal-mode growth, somewhat akin to a laser growth mechanism. Corresponding author address: Nili Harnik, Dept. of Geophys- ics and Planetary Sciences, Tel Aviv University, P.O. Box 39040, Since this theory is based on quite general wave prop- Tel Aviv 69978, Israel. erties, it deals both with gravity wave and vorticity wave E-mail: [email protected] instabilities. A very different approach, which has been DOI: 10.1175/2007JAS2610.1 © 2008 American Meteorological Society Unauthenticated | Downloaded 09/28/21 09:27 PM UTC JAS2610 2616 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 developed for Rossby waves, is based on the notion of associated with the “divergent” part of the flow, are counterpropagating Rossby waves1 (CRWs; Bretherton filtered out. From the gravity wave perspective, how- 1966; Hoskins et al. 1985). Viewed this way, instability ever, the dynamics can be nondivergent, when viewed arises from a mutual reinforcing and phase locking of in three dimensions. Moreover, as we show later on, such waves. This explicit formulation applies only to gravity waves involve vorticity dynamics, and this part Rossby waves. of the dynamics has the same action-at-a-distance fea- Recently, the seemingly different overreflection and tures as quasigeostrophic dynamics. CRW approaches to shear instability have been united This paper presents a vorticity–buoyancy view of for the case of Rossby waves. A generalized form of gravity waves, examines how this interplay between CRW theory describes the perturbation evolution in vorticity and buoyancy affects the evolution of strati- terms of kernel–wave interactions. Defining a local vor- fied shear flow anomalies, and explores its use as a basis ticity anomaly, along with its induced meridional veloc- for a kernel view. Our general motivation in developing ity, as a kernel Rossby wave (KRW), Heifetz and Meth- a vorticity-based kernel view of the dynamics is quite ven (2005, hereafter HM) wrote the PV evolution equa- basic: in a similar way in which the KRW perspective tion in terms of mutual interactions between the has yielded mechanistic understanding for barotropic KRWs, via a meridional advection of background PV, and baroclinic shear flows we expect that finding the in a way that was mathematically similar to the classical corresponding gravity wave building blocks will provide CRW formulation. The KRWs are kernels to the dy- a new fundamental insight into a variety of stratified namics in a way analogous to a Green function kernel. shear flow phenomena, such as a basic mechanistic Harnik and Heifetz (2007, hereafter HH07) used this (rather than mathematical) understanding of the cross- kernel formulation to show that KRW interactions are shear propagation of gravity wave signals, the necessary at the heart of cross-shear Rossby wave propagation conditions for instability, the overreflection mecha- and other basic components of overreflection theory, nism, and the nonmodal growth processes in energy and in particular, they showed that overreflection can and enstrophy norms. be explained as a mutual amplification of KRWs. The paper is structured as follows. After formulating Given that CRW theory can explain the basic com- the simplified stratified shear flow equations in terms of ponents of Rossby wave overreflection theory (e.g., vorticity–buoyancy dynamics (section 2), we examine wave propagation, evanescence, full, partial, and over- how the interaction between these two fields is manifest reflection) using its own building blocks (KRWs), and in normal modes in general (section 3a), in pure plane overreflection theory, in turn, can rationalize gravity waves (section 3b) and in a single interface between con- wave instabilities, it is natural to ask whether a wave– stant buoyancy and vorticity regions (section 3c). We then kernel interaction approach exists for gravity waves as go on to develop a kernel framework in section 4, for a well. Indeed, it has been shown that a mutual amplifi- single interface (section 4a), two and multiple interfaces cation of counter-propagating waves applies also to the (sections 4b and 4c), and the continuous limit (section interaction of Rossby and gravity waves, or to mixed 4d). We discuss the results and summarize in section 5. vorticity–gravity waves (Baines and Mitsudera 1994; Sakai 1989). These studies did not explicitly discuss the 2. General formulation case of pure gravity waves, in the absence of back- We consider an inviscid, incompressible, Boussinesq, ground vorticity gradients. In this paper we present a 2D flow in the zonal–vertical (x–z) plane, with a zonally general formulation of the dynamics of linear stratified uniform basic state that varies with height and is in shear flow anomalies in terms of a mutual interaction of hydrostatic balance. analogous kernel gravity waves (KGWs). We show how We start with the momentum and continuity equa- this formulation holds even when vorticity gradients, tions, linearized around this basic state: and hence “Rossby-type” dynamics, are absent. Du 1 Ѩp At first glance, Rossby waves and gravity waves seem ϭϪ Ϫ ͑ ͒ wUz ␳ Ѩ , 1a to involve entirely different dynamics. The common Dt 0 x framework used to describe Rossby waves is the quasi- Dw 1 Ѩp ϭ Ϫ ͑ ͒ b ␳ Ѩ , 1b geostrophic (QG) one, in which motions are quasi- Dt 0 z horizontal. In this framework, gravity waves, which are Db ϭϪwN 2, ͑1c͒ Dt 1 Since Rossby waves propagate in a specific direction, set by Ѩu Ѩw the sign of the mean PV gradient, their propagation is either with ϩ ϭ 0, ͑1d͒ or counter the zonal mean flow. Ѩx Ѩz Unauthenticated | Downloaded 09/28/21 09:27 PM UTC AUGUST 2008 HARNIK ET AL. 2617 x), u ϭ (u, w)isthe ity and displacement perturbations, we can determineץ/ץ)t) ϩ Uץ/ץ) where (D/Dt) ϵ perturbation velocity vector and its components in the the vertical velocity associated with the vorticity per- zonal and vertical directions, respectively; U and Uz are turbation. We note that the vertical velocity is nonlocal, the zonal mean flow and its vertical shear, respectively; in the sense that it depends on the entire vorticity p is the perturbation pressure; ␳␱ is a constant reference anomaly field. Given the vertical velocity field, it will ϵϪ␳ ␳ density; b ( / 0)g is the perturbation buoyancy; locally determine how the displacement field evolves, ץ ␳ץ ϵϪ ␳ 2 N (g/ 0)( / z) is the mean flow Brunt–Väisälä and along with the displacement field, will also deter- frequency, with ␳ and ␳ as the perturbation and mean mine how the vorticity field evolves. This scheme of flow density, respectively; and g is gravity. We note that inverting the vorticity field to get a vertical velocity (via 2 ϭ N bz, and use this in further notation. a streamfunction), then using this velocity to time inte- We now take the curl of Eqs.
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